Methods and systems for predicting sensitivity of blood flow calculations to changes in anatomical geometry

ABSTRACT

Embodiments include methods and systems and for determining a sensitivity of a patient&#39;s blood flow characteristic to anatomical or geometrical uncertainty. For each of one or more of individuals, a sensitivity of a blood flow characteristic may be obtained for one or more uncertain parameters. An algorithm may be trained based on the sensitivities of the blood flow characteristic and one or more of the uncertain parameters for each of the plurality of individuals. A geometric model, a blood flow characteristic, and one or more of the uncertain parameters of at least part of the patient&#39;s vascular system may be obtained for a patient. The sensitivity of the patient&#39;s blood flow characteristic to one or more of the uncertain parameters may be calculated by executing the algorithm on the blood flow characteristic of at least part of the patient&#39;s vascular system, and one or more of the uncertain parameters.

RELATED APPLICATION

This application claims the benefit of priority from U.S. ProvisionalApplication No. 61/948,325, filed Mar. 5, 2014, which is herebyincorporated herein by reference in its entirety.

TECHNICAL FIELD

Embodiments of the present disclosure relate to methods and systems forpatient-specific modeling of blood flow and, more particularly, tomethods and systems for sensitivity analysis in patient-specificmodeling of blood flow.

BACKGROUND

Coronary artery disease is a very common ailment that affects millionsof people annually. While significant strides have been made in thetreatment of coronary artery disease, including the use of percutaneouscoronary intervention (PCI) and coronary artery bypass graft surgery(CABG), a good understanding of the pathogenesis and the mechanism ofthe disease is still lacking.

Coronary artery disease may cause the blood vessels providing blood tothe heart to develop lesions, such as a stenosis (abnormal narrowing ofa blood vessel). As a result, blood flow to the heart may be restricted.A patient suffering from coronary artery disease may experience chestpain, referred to as chronic stable angina during physical exertion orunstable angina when the patient is at rest. A more severe manifestationof disease may lead to myocardial infarction, or heart attack.

A need exists to provide more accurate data relating to coronarylesions, for example, size, shape, location, functional significance(e.g., whether the lesion impacts blood flow), etc. Patients sufferingfrom chest pain and/or exhibiting symptoms of coronary artery diseasemay be subjected to one or more tests that may provide some indirectevidence relating to coronary lesions. For example, noninvasive testsmay include electrocardiograms, biomarker evaluation from blood tests,treadmill tests, echocardiography, single positron emission computedtomography (SPECT), and positron emission tomography (PET). Thesenoninvasive tests, however, typically do not provide a direct assessmentof coronary lesions or assess blood flow rates. The noninvasive testsmay provide indirect evidence of coronary lesions by looking for changesin electrical activity of the heart (e.g., using electrocardiography(ECG)), motion of the myocardium (e.g., using stress echocardiography),perfusion of the myocardium (e.g., using PET or SPECT), or metabolicchanges (e.g., using biomarkers).

For example, anatomic data may be obtained noninvasively using coronarycomputed tomographic angiography (CCTA). CCTA may be used for imaging ofpatients with chest pain and involves using computed tomography (CT)technology to image the heart and the coronary arteries following anintravenous infusion of a contrast agent. However, CCTA also cannotprovide direct information on the functional significance of coronarylesions, for example, whether the lesions affect blood flow. Inaddition, since CCTA is purely a diagnostic test, it can neither be usedto predict changes in coronary blood flow, pressure, or myocardialperfusion under other physiologic states (e.g., exercise).

Thus, patients may require an invasive test, such as diagnostic cardiaccatheterization, to visualize coronary lesions. Diagnostic cardiaccatheterization may include performing conventional coronary angiography(CCA) to gather anatomic data on coronary lesions by providing a doctorwith an image of the size and shape of the arteries. CCA, however, doesnot provide data for assessing the functional significance of coronarylesions. For example, a doctor may not be able to diagnose whether acoronary lesion is harmful without determining whether the lesion isfunctionally significant. Thus, CCA has led to a procedure referred toas an “oculostenotic reflex,” in which interventional cardiologistsinsert a stent for every lesion found with CCA regardless of whether thelesion is functionally significant. As a result, CCA may lead tounnecessary operations on the patient, which may pose added risks topatients and may result in unnecessary heath care costs for patients.

During diagnostic cardiac catheterization, the functional significanceof a coronary lesion may be assessed invasively by measuring thefractional flow reserve (FFR) of an observed lesion. FFR may be definedas the ratio of the mean blood pressure or flow downstream of a lesiondivided by the mean blood pressure or flow upstream from the lesion, forexample, the aortic pressure, under conditions of increased coronaryblood flow, for example, when induced by intravenous administration ofadenosine. Blood pressures may be measured by inserting a pressure wireinto the patient. Thus, the decision to treat a lesion based on thedetermined FFR may be made after the initial cost and risk of diagnosticcardiac catheterization has already been incurred.

To reduce the above disadvantages of invasive FFR measurements,HeartFlow Inc. has developed methods for assessing coronary anatomy,myocardial perfusion, and coronary artery flow noninvasively.Specifically, computational fluid dynamics (CFD) simulations have beensuccessfully used to predict spatial and temporal variations of flowrate and pressure of blood in arteries, including FFR. Such methods andsystems benefit cardiologists who diagnose and plan treatments forpatients with suspected coronary artery disease, and predict coronaryartery flow and myocardial perfusion under conditions that cannot bedirectly measured, for example, exercise, and to predict outcomes ofmedical, interventional, and surgical treatments on coronary arteryblood flow and myocardial perfusion.

However, computational modeling of hemodynamics involves a reconstructedgeometry of the patients' arteries, which is facilitated throughhigh-resolution imaging. For example, many CFD frameworks fornoninvasively calculating FFR assume that: (i) the geometry is knownwith certainty, (ii) clinical variables such as blood pressure,hematocrit, myocardial mass, etc. are known with certainty, and/or (iii)boundary conditions at the inlet and outlets of the computational modelare known with certainty. However, in reality, the FFR predicted usingCFD varies based on the accuracy of the available data and mathematicalmodels that describe hemodynamics in the arteries. As a result, there isa need for methods and systems for incorporating and quantifying theeffects of uncertainties in the available data, as well as mathematicalmodels. In addition, there is a need for methods and systems forassigning confidence intervals to determined FFR values as well asranking the sensitivity of FFR calculation to the possible values ofdifferent parameters.

The foregoing general description and the following detailed descriptionare exemplary and explanatory only and are not restrictive of thedisclosure.

SUMMARY

Embodiments discussed herein broadly concern predicting sensitivities ofblood flow calculations to changes in an anatomical geometry. A digitalrepresentation, such as a scan of a patient's heart, often containsimperfections, or uncertainties. These uncertainties may affect variousresulting blood flow calculations to a great degree, or possibly notmuch at all. The degree to which uncertainties in a digitalrepresentation, such as a heart scan, affect resulting blood flowcalculations may be called the sensitivity. For example, if a scan of apatient's heart contains an artifact such as misalignment, motion,blooming, etc., and that artifact results in large errors in blood flowcalculations that are made based on the scan, then the sensitivity ishigh. As discussed in various embodiments herein, machine learning andother algorithms may be used to predict various sensitivities, and hencethe confidence a physician or technician may place in blood flowcalculations for a given patient.

In accordance with an embodiment, methods are disclosed for predictinggeometric sensitivity, using at least one computer system. One methoddetermines a sensitivity of a patient's blood flow characteristic toanatomical or geometrical uncertainty, the method comprising: obtaining,for each of a plurality of individuals, a sensitivity of a blood flowcharacteristic to one or more uncertain parameters; training amachine-learning algorithm based on the sensitivities of the blood flowcharacteristic and one or more of the uncertain parameters for each ofthe plurality of individuals; obtaining, for a patient, a geometricmodel, a blood flow characteristic, and one or more of the uncertainparameters of at least part of the patient's vascular system; andcalculating a sensitivity of the patient's blood flow characteristic toone or more of the uncertain parameters by executing themachine-learning algorithm on the blood flow characteristic of at leastpart of the patient's vascular system, and one or more of the uncertainparameters.

In accordance with another embodiment, a method is disclosed forpredicting geometric sensitivity, using at least one computer system,the method comprising: receiving a first digital representation of afirst physical system; determining at least one uncertainty of at leastone property of the first digital representation; calculating at leastone sensitivity for the uncertainty using a stochastic algorithm;determining a machine learning predictor using data from a plurality ofpatients; and predicting sensitivities for a second digitalrepresentation of a second physical system using the machine learningpredictor.

In accordance with another embodiment, a system is disclosed forpredicting geometric sensitivity, the system comprising: a data storagedevice storing instructions for predicting geometric sensitivity; and aprocessor configured to execute the instructions to perform a methodincluding the steps of: receiving a first digital representation of afirst physical system; determining at least one uncertainty of at leastone property of the first digital representation; calculating at leastone sensitivity for the uncertainty using a stochastic algorithm;determining a machine learning predictor using the at least onesensitivity; and predicting sensitivities for a second digitalrepresentation of a second physical system using the machine learningpredictor.

In accordance with another embodiment, a non-transitory computerreadable medium for use on at least one computer system containingcomputer-executable programming instructions for predicting geometricsensitivity, the method comprising: receiving a first digitalrepresentation of a first physical system; determining at least oneuncertainty of at least one property of the first digitalrepresentation; calculating at least one sensitivity for the uncertaintyusing a stochastic algorithm; determining a machine learning predictorusing the at least one sensitivity; and predicting sensitivities for asecond digital representation of a second physical system using themachine learning predictor.

Additional objects and advantages of the disclosed embodiments will beset forth in part in the description that follows, and in part will beapparent from the description, or may be learned by practice of thedisclosed embodiments. The objects and advantages of the disclosedembodiments will be realized and attained by means of the elements andcombinations particularly pointed out in the appended claims.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory onlyand are not restrictive of the disclosed embodiments, as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate several embodiments and togetherwith the description, serve to explain the principles of the disclosure.

FIG. 1 is a block diagram of an exemplary system and network forpredicting sensitivity of blood flow calculations to changes inanatomical geometry, according to an exemplary embodiment of the presentdisclosure;

FIG. 2 is a flow chart that describes an exemplary process for geometricsensitivity prediction, according to an exemplary embodiment of thepresent disclosure;

FIG. 3 is a flow chart that describes an exemplary process forsensitivity analysis in patient-specific modeling of blood flow,according to an exemplary embodiment of the present disclosure;

FIG. 4 is a flow chart that describes an embodiment for sensitivity oflift and drag forces to airfoil shape, according to an exemplaryembodiment of the present disclosure;

FIG. 5 is a flow chart that describes an exemplary process forsensitivity prediction of growth and remodeling response to the geometryof a stented artery, according to an exemplary embodiment of the presentdisclosure;

FIG. 6 is a diagram illustrating an exemplary sensitivity map calculatedusing machine learning, according to an exemplary embodiment of thepresent disclosure; and

FIG. 7 is a flow chart that describes an exemplary process for geometricsensitivity prediction, according to an exemplary embodiment of thepresent disclosure.

DESCRIPTION OF THE EMBODIMENTS

Reference will now be made in detail to the exemplary embodiments of theinvention, examples of which are illustrated in the accompanyingdrawings.

In an exemplary embodiment, a method and system determines informationrelating to blood flow in a specific patient using information retrievedfrom the patient noninvasively. Various embodiments of such a method andsystem are described in greater detail in U.S. Pat. No. 8,315,812 (“the'812 patent”), filed Jan. 25, 2011, and entitled “Method and System forPatient-Specific Modeling of Blood Flow,” which is hereby incorporatedherein by reference in its entirety.

In some embodiments, the information determined by the method and systemmay relate to blood flow in the patient's coronary vasculature.Alternatively, the determined information may relate to blood flow inother areas of the patient's vasculature, such as carotid, peripheral,abdominal, renal, and cerebral vasculature. The coronary vasculatureincludes a complex network of vessels ranging from large arteries toarterioles, capillaries, venules, veins, etc. The coronary vasculaturecirculates blood to and within the heart and includes an aorta thatsupplies blood to a plurality of main coronary arteries (e.g., the leftanterior descending (LAD) artery, the left circumflex (LCX) artery, theright coronary (RCA) artery, etc.), which may further divide intobranches of arteries or other types of vessels downstream from the aortaand the main coronary arteries. Thus, the exemplary method and systemmay determine information relating to blood flow within the aorta, themain coronary arteries, and/or other coronary arteries or vesselsdownstream from the main coronary arteries. Although the aorta andcoronary arteries (and the branches that extend therefrom) are discussedbelow, the disclosed method and system may also apply to other types ofvessels.

In an exemplary embodiment, the information determined by the disclosedmethods and systems may include, but is not limited to, various bloodflow characteristics or parameters, such as blood flow velocity,pressure (or a ratio thereof), flow rate, and FFR at various locationsin the aorta, the main coronary arteries, and/or other coronary arteriesor vessels downstream from the main coronary arteries. This informationmay be used to determine whether a lesion is functionally significantand/or whether to treat the lesion. This information may be determinedusing information obtained noninvasively from the patient. As a result,the decision whether to treat a lesion may be made without the cost andrisk associated with invasive procedures.

As described above, the above-described techniques for computationalmodeling for noninvasively calculating FFR may assume that: (i) thegeometry is known with certainty, (ii) clinical variables such as bloodpressure, hematocrit, myocardial mass, etc. are known with certainty,and/or (iii) boundary conditions at the inlet and outlets of thecomputational model are known with certainty. However, in reality, theFFR predicted using CFD varies based on the accuracy of the availabledata and mathematical models that describe hemodynamics in the arteries.Accordingly, the present disclosure also describes methods and systemsfor incorporating and quantifying the effects of uncertainties in theavailable data, as well as mathematical models. In addition, the presentdisclosure describes methods and systems for determining FFR values, aswell as rank the sensitivity of FFR calculations to differentparameters. In one embodiment, the present disclosure incorporates thesensitivity analysis techniques disclosed in U.S. patent applicationSer. No. 13/864,996 (“the '996 application”), filed on Apr. 17, 2013,the entire disclosure of which is incorporated herein by reference. Inanother embodiment, the present disclosure incorporates the sensitivityanalysis techniques disclosed in U.S. patent application Ser. No.14/231,870 (“the '870 application”), filed on Apr. 1, 2014, the entiredisclosure of which is incorporated herein by reference.

In contrast to and addition to the '996 application, the presentdisclosure describes various methods for using machine learning andother algorithmic techniques for performing sensitivity analysis,including defining input uncertainties, and calculating FFR analysissensitivities, according to a general and several specific exemplaryembodiments. Sensitivities of determined blood flow characteristics,such as FFR, to uncertain parameters, uncertain clinical variables,and/or uncertain geometries may thus be determined more quickly byrelying on machine learning techniques trained based on relatively largedatabases or data sets of training data of patient geometricsensitivity.

It should be appreciated that blood flow modeling in the coronaryarteries may be performed by solving the Navier-Stokes equations, whichare partial differential equations describing blood velocity andpressure. Computational fluid dynamics (CFD) simulations may be used topredict spatial and temporal variations of flow rate and pressure ofblood in arteries, including FFR. CFD simulations may rely on (i) anaccurate reconstruction of the coronary artery geometry from a CT scan,(ii) an accurate representation of the system properties such as bloodviscosity, aortic pressure, flow-split across different arteries, etc.,and (iii) accurate enforcement of boundary conditions, which describethe properties of a microvascular coronary bed that is not modeled. Inpractice, it may not be possible to obtain the properties mentionedabove with certainty. Hence, it may be useful to quantify theuncertainty, for example by calculating the standard deviation andhigher statistical moments including confidence intervals of FFR.

Stochastic Navier-Stokes equations may be used to solve for many typesof velocity and pressure variables that depend on a stochastic, orprobabilistic, dimension, in addition to space and time. Thecomputational complexity depends on the problem and the number ofsources of uncertainties that are accounted for. If the reconstructedgeometry is stochastic, then the number of stochastic dimensions couldeasily be of the order of 50 to 100. Since solving the deterministicNavier-Stokes equations themselves is computationally intensive andcould involve significant processing time depending on the size of thecoronary tree, traditional methods, such as Monte-Carlo, can beinefficient and impractical to use. Classes of techniques called“stochastic collocation methods” may be highly efficient in quantifyinguncertainty. However, they still require hundreds of simulationstranslating to significant computational Stochastic effort.

Embodiments disclosed herein may include using a machine learningpredictor as a surrogate for calculating FFR. This may be achieved bydefining patient-specific features, and computing a map between thesefeatures and FFR by training on a large database of patients for whichCFD results have been computed. An extensive set of features may bechosen encompassing geometric, clinical, and analytical model basedfeatures, such as analytical solutions for pipe flow parameterized bypipe radii, flow rate, viscosity, and the length of the pipe. This makesthe uncertainty quantification problem tractable, reducing thecomputational time for the stochastic algorithm to approximately fromdays to under an hour. However, it is desirable to reduce this timefurther so that real time computation of sensitivity is possible.

More specifically, the techniques presented herein may utilize adatabase of patient-specific sensitivities to construct a machinelearning predictor for calculating sensitivities. This machine learningpredictor may encode the stochastic collocation algorithm in addition tothe Navier-Stokes equation. In addition to the features used forcalculating FFR (discussed in the '812 patent and '996 applicationreferenced and incorporated herein above), the following additionalfeatures may be added: (i) magnitude of input uncertainty; (ii) FFRcalculated using machine learning; and (iii) geometry parameterization.In one embodiment, FFR may be calculated using machine learningaccording to any of the techniques disclosed in U.S. patent applicationSer. No. 13/895,893, filed May 16, 2013, by Grady, Choi, and Singer, andSer. No. 13/895,871, filed May 16, 2013, by Fonte, Choi, Grady, andSinger, the entire disclosures of which are hereby incorporated hereinby reference.

Since geometry parameterization may be used as an input feature,arbitrary spatial resolution in sensitivity can be achieved with almostno additional cost. This method can be used for calculating sensitivityto geometry in other applications, such as sensitivity of drag and liftforces to airfoil shape, sensitivity of wall shear stress to stentdeployment geometry, sensitivity of drilling location to uncertainty indispersion index using flow simulations in porous media, etc.

Referring now to the figures, FIG. 1 depicts a block diagram of aexemplary methods and systems for predicting sensitivity of blood flowcalculations to changes in anatomical geometry. Specifically, FIG. 1depicts a plurality of physicians 102 and third party providers 104, anyof whom may be connected to an electronic network 100, such as theInternet, through one or more computers, servers, and/or handheld mobiledevices. Physicians 102 and/or third party providers 104 may create orotherwise obtain images of at least a portion of the anatomy of aplurality of individuals, and of at least one patient. For example,physicians 102 and/or third party providers 104 may create or otherwiseobtain images of at least a portion of one or more individuals' and/orpatients' cardiac and/or vascular systems. The physicians 102 and/orthird party providers 104 may also obtain any combination ofpatient-specific information, such as age, medical history, bloodpressure, blood viscosity, etc. Physicians 102 and/or third partyproviders 104 may transmit the cardiac/vascular images and/orpatient-specific information to server systems 106 over the electronicnetwork 100. Server systems 106 may include storage devices for storingimages and data received from physicians 102 and/or third partyproviders 104. Server systems 106 may also include processing devicesfor processing images and data stored in the storage devices.

An exemplary general embodiment will next be described, with referenceto FIG. 2, followed by several specific embodiments. As shown in theembodiment 200, sources of uncertainties associated with a digitalrepresentation may be determined, and machine learning, or otheralgorithms, such as lookup tables, may be used to predict one or moresensitivities.

As shown in step 205, a digital representation of the system may firstbe acquired, for example, of a system that is to be studied. This couldinclude image-based representation, measured variables, a list, or atable of parameter values and features representative of the system, ora combination of the above. The system may then be isolated to bestudied. This may be done by delineating the geometry, systemproperties, and specific conditions. This could include additional stepssuch as image processing and reconstructing the system from a rawdigital representation.

Sources of uncertainties may then be defined and modeled, as shown instep 210. Uncertainties in the geometry, system properties, and/orconditions imposed, which are relevant to the quantities of interest,may be defined. Uncertainty can be associated with an entity based on(a) inherent noise in the measurement or acquisition of the entity, (b)uncertainty in the algorithm to delineate the entity from digitalrepresentation, and/or (c) uncertainty due to the discreterepresentation of a dynamic system (e.g., using a finite set of staticsnapshots to represent the system). A form of probability distributionfunction (pdf) may be assigned to each of the entities defined above.The form of pdf could be motivated by understanding the cause and sourceof uncertainty, empirical data, and/or physical reasoning. Commonly usedpdfs are Gaussian and Uniform distributions.

Sensitivities may be calculated using machine learning, as shown in step215. The geometry may be parameterized by splitting the geometry of thesystem into multiple regions. Each such split may be eventually assigneda sensitivity value. Correlations between different regions in thegeometry can also be modeled. This split can be performed either usingsalient locations of the system which encode different physicalproperties and/or features of the system, and/or simply to achieve adesired resolution in sensitivity. A stochastic algorithm may also beinitialized. An adaptive stochastic collocation algorithm may beutilized to compute specific instances of entities where simulations maybe performed to calculate the quantity of interest. For example, thestochastic collocation algorithm may compute instances of geometry froma probabilistic space of geometries using quadrature points identifiedby the Smolyak sparse grid algorithm. Sensitivities may also becalculated by solving the stochastic system. Sensitivities may becalculated by running the simulations at quadrature points identified induring the initialization of the stochastic algorithm. These simulationscan be run by either using computational methods by discretizing thegeometry and solving partial differential equations (e.g. finite elementmethods), or by using surrogate functions such as the response surfacemethod, reduced order models, machine learning, etc.

As shown in step 220, a machine-learning predictor may also beconstructed for sensitivities. Available data may be split into atraining and test set. The dataset may be split where sensitivities werecalculated, by solving the stochastic system, into a training and testset. A random split of 66% training set and 34% test set may be useful,though other pre-determined splits may also be used.

While in training mode, features may be defined and calculated.Problem-specific features that are important in the calculation ofsensitivities may be defined. These include (a) problem-specificgeometric features, such as spatial variation in area and diameter,branching features (location/size of nearest branch), volumetricfeatures (e.g. total volume upstream or downstream of a given locationfor flow simulations); (b) problem-specific variables or parameters,such as viscosity or density (for flow problems), constitutive materialproperties for solid mechanics problems, dispersivity for flow in porousmedia etc.; and/or (c) stochastic variables, such as the magnitude ofinput uncertainty, the form of input uncertainty, and theparameterization of the geometry into independent random variables(number of stochastic dimensions). Mathematical transformation of thefeatures may be performed. For example, mathematical transformations,such as squaring, can improve the separation of feature values andhence, prediction error. Optimal regressor(s) may also be calculated.One or more optimal machine learning regressors may be determined fromat least one regressor candidate. These could include linear regressionmodels, non-linear polynomial models, decision trees, M5 rules (terminalleaves of decision trees have a linear regression model), etc. Bootstrapaggregating may also be used to improve performance. The regressor mayalso be saved to a digital representation (e.g., the memory or digitalstorage [e.g., hard drive, network drive] of a computational device suchas a computer, laptop, DSP, server, etc.).

In a production mode, data may be obtained for at least one newrepresentation of the system. The saved regressor from training may beto calculate sensitivities for the new digital representation.

A specific exemplary embodiment of the present disclosure forsensitivity analysis in patient-specific modeling of blood flow will nowbe discussed, with reference to FIG. 3. As shown in the embodiment 300,sources of uncertainties associated with a digital representation may bedetermined, and machine learning, or other algorithms, may be used topredict one or more sensitivities for the current patient.

As shown in step 305, for one or more patients, a digital representationmay be acquired (e.g., the memory or digital storage [e.g., hard drive,network drive] of a computational device such as a computer, laptop,DSP, server, etc.) and processed. The digital representation mayinclude, for example, an image scan of the patient that includes theascending aorta and/or coronary artery tree. The type of image scancould include a cardiac computed tomography (CCTA), MRI, ultrasound etc.Using the image scan, a digital representation that encompasses regionsof interest may be isolated. Centerlines, which pass through the centerof vessels of interest, may be computed. Subsequently, these may be usedto construct lumen segments manually or automatically to identify voxelsbelonging to the aorta and to the lumen of the coronary arteries. Onceall relevant voxels are identified, a geometric model of the aorta andrelevant coronary arteries may be reconstructed. In addition to a CCTA,etc., a set of clinical parameters may be measured or obtained whichinclude heart-rate, systolic and diastolic brachial blood pressures,hematocrit, patient height and weight, and patient history such assmoking status, presence/absence of diabetes, etc.

Based upon the image scan and the reconstructed geometric model, a setof derived quantities may be calculated, as shown in step 310. Thesederived quantities may include: (a) myocardial mass (m_(myo)), which maybe obtained by image segmentation of the left ventricle to calculate thevolume of myocardium and multiplying it with a blood density; (b) bodysurface area, which may be calculated from the patient height (h) andweight (w) as

${{BSA} = \sqrt{\frac{hw}{3600}}};$

(c) viscosity, which may be calculated from the hematocrit (hem) as

$\eta = \frac{c}{\left( {1 - \frac{hem}{100}} \right)^{2.5}}$

where c may be 0.0012; (d) inlet aortic flow rate (Q), which may becalculated from scaling studies as

${Q = {\frac{1}{60}{BSA}^{1.15}}};$

(e) coronary flow rate (q_(cor)), which may be calculated frommyocardial mass as

$q_{cor} = {c_{dil}\frac{5.09}{60}m_{myo}^{0.75}}$

where c_(dil) is the dilation factor; (f) coronary resistance, where thenet coronary resistance may be calculated from the desired coronaryflow, and the value for individual outlets is calculated based on theirareas; and/or (g) resistance of outlet aorta, which may be calculatedbased on aortic pressure, aortic flow rate and/or desired coronary flowrate, for example.

For each patient that underwent CCTA, the input uncertainties may beidentified and modeled, as shown in step 315. Derived quantities such asclinical parameters from step 310 may be modeled using (i) uniformprobability distribution function if only the range of the parameter isknown, and/or (ii) Gaussian probability distribution function, if a meanand standard deviation are known. Geometry variables, such as thesurface of the geometry, and consequently lumen area and wall boundariesmay also be considered uncertain. The number of stochastic geometricvariables may depend on how the geometry is discretized. The trainingand test modes may be discretized differently. In the training mode,each segment between two branch locations may be considered anindependent random variable. In the test mode, the discretization couldbe made as fine as possible, such as, for example, every set of sixcenterline points can be considered an independent variable.

Sensitivity data may be computed using machine learning on FFR, as shownin step 320. Sensitivity data may be calculated on an online and/oroffline database of patient geometries. For each patient-specific datain the database, one or more geometric features, features from clinicaldata, and/or hemodynamic features may be calculated. Geometric featuresmay include calculating or obtaining lumen area, lumen diameter, volumeof blood upstream and downstream, distance to nearest bifurcationupstream and downstream, distance to ostium (location where aorta meetscoronary artery), min/max/mean upstream and downstream diameter,distance to minimum upstream and downstream diameter, number of upstreamand downstream bifurcations, average upstream and downstream diameter,min/max/mean area of all downstream outlets, min/max/mean outletresistances, area of next centerline point, and/or total and meandownstream geometric resistance. Features from clinical data may includecalculating or obtaining the viscosity of blood, systolic and diastolicpressures, body surface area, and/or estimates of aortic and coronaryblood flow rate. Hemodynamic features may also be calculated orobtained, which may include flow rates, pressure drops, and FFR usingPoiseulle's equation, kinetic energy drop, and/or pressure recoveryfactor (which may be defined as the ratio of downstream to upstream areaif this ratio is >1.2, although this pre-determined ratio may vary).

FFR values may be calculated by performing 3D blood flow simulations bysolving Navier-Stokes equations for patients in a database, such as anoffline database.

A machine-learning predictor may be calculated for FFR. The database maybe divided into, for example, a 66% training set and 34% test set. Thispre-determined division may vary. The method may further includecomputing a decision tree that maps the features previously identifiedin step 320 and the simulation results, so that predictive error isminimized. The resulting predictor may be stored in digital form (e.g.,disc, laptop, hard-drive, etc.).

Sensitivities may also be calculated. Quadrature points may becalculated for performing uncertainty quantification using the Smolyakalgorithm. Each quadrature point to a feature set listed above may bemapped. For each quadrature point, FFR may be evaluated using a machinelearning predictor previously calculated. Lagrange polynomialinterpolates may be used to interpolate FFR in the stochastic space, andby sampling the stochastic space and constructing, for example, ahistogram, the standard deviation in FFR (sensitivity) may becalculated.

As shown in step 325, a machine-learning predictor for sensitivity mayalso be calculated (e.g., using one or more offline databases). Afeature vector, from step 320, may be enriched. The following featuresmay be used to predict sensitivity in addition to those listed above inregards to step 320—(i) length of independent geometric segments, (ii)magnitude of input uncertainty, and/or (iii) FFR calculated usingmachine learning, as well as non-linear transformations of the abovesuch as square, square root, log, etc. A machine-learning predictor maybe calculated. The one or more databases may be divided into apre-determined ratio, for example, a 66% training set and 34% test set,although this division may vary. A decision tree may also be computedthat maps the features identified in the enriched feature vector to thesensitivities calculated in step 320, so that predictive error isminimized. The resulting predictor may be stored in digital form (e.g.disc, laptop, hard-drive, etc.).

As shown in step 330, sensitivities may be predicted for at least onecurrent patient. Sensitivities for the patient data stored in step 305and 310 may be predicted by first calculating all the featuresidentified in step 325, and using the stored machine learning predictor,from step 325, to calculate sensitivities through the patient specificmodel. A multi-resolution analysis may also be performed. Desiredspatial resolution may be achieved by defining segment size based on thedesired resolution. Six centerline points may be chosen to define asegment, for example. Hence, a spatial resolution of 3 mm could beachieved if the distance between centerline points is 0.5 mm (see FIG.6).

An exemplary embodiment of the method to calculate the sensitivity oflift and drag forces to airfoil shape is now described, with referenceto FIG. 4. This has implications in the predictive modeling of drag andlift variability due to airfoil shape, as well as to the optimal designof aircraft wings. As shown in the embodiment 400, sources ofuncertainties associated with a digital representation or one or moreairfoils may be determined, and machine learning, or other algorithms,may be used to predict one or more sensitivities for the airfoil data.

As shown in step 405, input data may be acquired and processed. Theinput data may comprise the shape of the airfoil to be studied. Thiscould be in the form of an analytical function, free form representationusing a set of control points or some other representation such as aBezier or B-Spline control point representation. The quantity ofinterest may also be acquired and analyzed. This could be overall liftand drag forces on the aircraft, and/or localized forces on specificpoints in the aircraft wing, and/or other forces on the aircraft.

As shown in step 410, sources of uncertainties may be defined andmodeled. The model of uncertainty may be split into uncertainty inoperating conditions and uncertainty in shape. Regarding uncertainty inoperating conditions, aircrafts operate under widely varying conditionssuch as air temperature, density and head or tail winds, and/or loadingconditions. The goal is to operate safely and efficiently under a widerange of these conditions. The roll and yaw of aircraft could also betemporarily compromised or vary during flight. A Gaussian distributionin the operating conditions may be assumed, and nominal temperature andabsence of head or tail winds on average may also be assumed. Regardinguncertainty in shape, the shape of an aircraft wing can be uncertain dueto uncertainty in the manufacturing process or large deformation underunexpected operating conditions. Sensitivity to shape is also a usefulmetric when performing shape optimization for airfoil design. Astochastic space of candidate shapes may be determined.

The sensitivity may be calculated, for example, using an online and/oroffline database, as shown in step 415. A database containing a largenumber of airfoil shapes may be obtained. This may encompass airfoilswith different shapes. When running one or more simulations, lift anddrag forces on the airfoil may be calculated by solving Navier-Stokesequations for air velocity and pressure near the wall of the airfoil. Noslip boundary conditions may be enforced at the wall of the airfoil. Acomputational mesh of the air surrounding the airfoil may be created,with a boundary layer mesh surrounding the air foil to account for steepchanges in air velocity and pressure. The sensitivity may also becalculated using adjoint equations.

Analytical equations for sensitivities may be computed by derivingadjoint equations from governing Navier-Stokes equations. Theseequations are usually solved backwards in time, and have a heavycomputational demand (may need to store the history of the solution atall times) but are feasible to calculate in some situations. Solutionsof these equations provide information about sensitivity at all pointsin space. When it is not feasible to solve adjoint equations, astochastic collocation method can be used.

Further, the sensitivity may be calculated using stochastic collocation.A stochastic space representing uncertainty in operating conditions andshape may be defined. Navier-Stokes equations from above may be solvedat quadrature points identified by the stochastic collocation algorithm.The velocity and pressure of air may be calculated throughout thestochastic space using Lagrange polynomial interpolation. A histogrammay be constructed by sampling the stochastic space according to theinput probability defined in step 410, from which sensitivity values arecalculated.

As shown in step 420, a machine learning predictor for sensitivity maybe calculated. Regarding input features, for each airfoil shape, the setof input features may be computed, such as the number of control points,shape coefficients, the magnitude of input uncertainty, and/or loadingconditions. The number of control points may be points that define thegeometry of the airfoil. The shape coefficients are, for example,coefficients in the analytical representation of the airfoil shape.These could be the coefficients of Bezier curves, or piecewisepolynomial coefficients. The magnitude of input uncertainty defined instep 415 may also be computed. The loading conditions may also becomputed, such as the weight of the aircraft, head and tail wind forces,and thrust generated by the aircraft.

The database may be divided into a 66% training set and 34% test set,although this pre-determined split may vary. A decision tree may becomputed, or a polynomial regressor that maps the input featuresidentified above to the sensitivities calculated in step 415, so thatpredictive error is minimized. The resulting predictor may be stored indigital form (e.g. disc, laptop, hard-drive, network storage, etc.).

The sensitivities for the current input may then be predicted, as shownin step 425. The sensitivities for the airfoil data stored in step 405may be predicted by first calculating all the features identified instep 420, and using the stored machine learning predictor from step 420to calculate sensitivities through the patient specific model. Thesolution may be mapped and displayed. The sensitivity of lift and dragforces may be mapped to shape coefficients, to the sensitivity withrespect to the geometry. Store the maximum sensitivity at eachdiscretized point in the geometry, and display the results in a displaydevice.

FIG. 5 is now discussed, which shows a flow chart describing anexemplary process for sensitivity prediction of growth and remodelingresponse to the geometry of a stented artery (post stent deployment).The stenting procedure affects near-wall hemodynamics including wallshear stress and intramural stress, which in turn affects propensity forre-stenosis. As shown in the embodiment 500, sources of uncertaintiesassociated with a digital representation may be determined, and machinelearning, or other algorithms, may be used to predict one or moresensitivities.

As shown in step 505, input data may be acquired and/or processed. AnMRI or CT scan, for example, may be acquired of a patient who is acandidate for a stenting operation including the aorta and arteries ofinterest encompassing the stenting location. One or more stentdeployment plans including location and intended shape of the arterypost-surgery may also be acquired and/or processed.

The sources of uncertainties may be defined and/or modeled, as shown instep 510. For example, uncertainty in the stent deployment geometry maybe defined and modeled. Due to uncertainty associated with the locationand shape of the stenting procedure, the lumen geometry post-surgery ofa stented vessel may be uncertain. An expected shape of the lumenpost-surgery may be calculated based on the stent deployment plan (thediameter to which the balloon is inflated and the precise location). Thestandard deviation in the stent location and diameter may be estimatedto be a fraction of the mean expected parameters, where the fraction ispatient and lesion specific (for e.g., ostial/non-ostial lesions).

Clinical parameters that are relevant to the simulations may be assumedto be uncertain with either a uniform or Gaussian distribution.Uncertainties for hemodynamic simulations include measured bloodpressure, body mass index, blood viscosity, and/or downstream perfusionindices if available. Some of these uncertainties for arterial growthand remodeling simulations may include (i) the constitutive materialproperty of the wall, (ii) the stretch at which components such aselastin and collagen are deposited, and/or (iii) gain parameters, whichdetermine the response of vessel wall to deviations in stress fromhomeostatic values.

As shown in step 515, the sensitivity may be calculated using at leastone online and/or offline database. This step may be performed for alarge database (say 100 stented geometries) so that they capturepatient-to-patient variability such as lesions in different locations,vessels of different caliber and length, etc.

There are typically two sets of governing equations that may be solvedin a coupled fashion. Blood flow rate and pressures in the stented lumenmay be calculated based on estimated stented geometry by solvingNavier-Stokes equations. Intramural and shear stresses may be estimatedfrom these, and may be used to predict growth and remodeling responseafter which they are run for an extended time, such as one or more days.The new geometry may be fed back to the Navier-Stokes equation fromwhich updated wall shear stress and intramural stresses are estimated,and so on. The simulation may be terminated after sufficient time(typically months or years) to estimate remodeled state and thepropensity for re-stenosis. Simulation of growth and remodeling may beperformed by solving quasi-static stress equilibrium equations, andaccounting for the addition and removal of material to the wall duringremodeling process. These simulations are computationally expensive, andmight take significant time to solve.

The sensitivity may also be calculated using stochastic collocation. Astochastic shape space may be defined that describes uncertainty instent deployment geometry. Specific instances of the stented geometrymay be identified at quadrature points of the stochastic space using thestochastic collocation algorithm. Following this, hemodynamic, growthand/or remodeling simulations may be performed at each collocation pointas described above. The wall shear stress and intramural stress aftersufficient time (a few years) may be calculated throughout thestochastic space using an interpolation such as a Lagrange polynomialinterpolation. A histogram may be constructed by sampling the stochasticspace according to the input probability defined in step 510, from whichsensitivity values may be calculated.

A machine learning predictor for sensitivity may be calculated, asdiscussed in step 520. Regarding input features, for each of the stentedgeometries, one or more of the following set of input features may becalculated: (i) parameters of the geometry, (ii) hemodynamic indices,(iii) the magnitude of input uncertainty, (iv) the material propertiesof the vessel wall, and/or (v) the parameters driving the dynamics ofgrowth and remodeling. Regarding the parameters of the geometry, thefollowing may be calculated: pre-stented and post-stented lumen area,lumen diameter, volume of blood upstream and downstream in thecomputational model, distance to nearest branch upstream and downstreamof the stented location, min/max/mean upstream and downstream diameter,distance to minimum upstream and downstream diameter, min/max/meanoutlet resistances, and geometric resistance using, for example, thePoiseulle equation.

Hemodynamic indices may also be calculated, such as flow rates, pressuredrops, FFR using Poiseulle's equation, and kinetic energy drop in thepre-stented geometry and planned stented configuration. A pressurerecovery factor may be calculated, and may be defined as the ratio ofdownstream to upstream area if this ratio is >1.2 (although thispre-determined ratio may vary) and included in the feature vector.

The magnitude of input uncertainty that was defined for each entity instep 515 may also be calculated. The material properties of the vesselwall such as collagen, elastin, and smooth muscle cell may also becalculated. Further, parameters driving dynamics of growth andremodeling, such as the half-life of different constituents, pre-stretchof deposited constituents, and gain parameters, defined as the ratiobetween the mass of constituent added and deviation of stress fromhomeostatic values, may also be calculated.

The database may be divided into a 66% training set and 34% test set,although this pre-determined ratio may vary. A decision tree or apolynomial regressor may be computed that maps the features identifiedin step 520 to the sensitivities calculated in step 515, so thatpredictive error is minimized. The resulting predictor may be stored indigital form (e.g. disc, laptop, hard-drive, network drive, etc.).

As discussed in step 525, the sensitivities for current input may bepredicted. The sensitivities for the data stored in step 505 may bepredicted by calculating all the features identified in step 520, andusing the stored machine learning predictor (step 520) to calculatesensitivities through the patient specific model.

The solution may be mapped and displayed. The sensitivity of thequantities of interest to geometry through the vessel of interest may bedisplayed on a device. This could be used to ensure the fidelity andaccuracy of stent deployment at highly sensitive regions.

FIG. 6. depicts a diagram illustrating an exemplary sensitivity map 600calculated using machine learning according to an exemplary embodimentof the present disclosure. As discussed above, centerline points may bechosen to define a segment. For example, in reference to the firstspecific embodiment presented above, a spatial resolution of 3 mm couldbe achieved if the distance between centerline points is 0.5 mm. Asensitivity map display 605 may be shown to a physician and/or user witha sensitivity scale 610 to allow a physician and/or user to interpretthe display. The sensitivity scale 610 may display colors or otherwiseprovide indicators as to the sensitivity values of different portions ofa structure such as, for example, a patient's coronary arteries. Colorsof other indicators 615 may be projected onto the display correspondingto different sensitivity values. This means that each color or otherindicator 615 may indicate the sensitivity of the blood flowcalculations to changes or uncertainty in the geometry. This isimportant and/or useful because the technician or physician knows whereto focus his or her efforts.

FIG. 7 is a flow chart that describes an exemplary process for geometricsensitivity prediction according to an exemplary embodiment of thepresent disclosure. At step 705, a first digital representation of afirst physical system is received. At step 710, at least one uncertaintyof at least one property of the first digital representation isdetermined. At step 715, at least one sensitivity is calculated for theuncertainty using a stochastic algorithm. At step 720, a machinelearning predictor utilizing a plurality of patients is determined. Atstep 725, sensitivities for a second digital representation of a secondphysical system using the machine learning predictor are predicted.

As a result of embodiments discussed herein, a technician and/orphysician may be able to predict the sensitivities of blood flow orother properties to uncertainty in an input digital representation.Other embodiments of the invention will be apparent to those skilled inthe art from consideration of the specification and practice of theinvention disclosed herein. It is intended that the specification andexamples be considered as exemplary only, with a true scope and spiritof the invention being indicated by the following claims.

1. A computer-implemented method of determining a sensitivity of apatient's blood flow characteristic to uncertainty in one or moreanatomical or geometrical features of the patient, the methodcomprising: obtaining, for each of a plurality of individuals, ageometric model and a sensitivity of a blood flow characteristic to atleast one uncertainty, wherein the at least one uncertainty comprisesone or more uncertain parameters, uncertain clinical variables, and/oruncertain geometries; obtaining, for a patient, a geometric model of atleast part of the patient's vascular system, a designation of a bloodflow characteristic, and a value of the at least one uncertainty for atleast part of the patient's vascular system; and determining asensitivity of the blood flow characteristic of the patient for at leastthe part of the patient's vascular system, based on one or moresensitivities of the plurality of individuals.
 2. The method of claim 1,wherein obtaining a sensitivity of a blood flow characteristic to the atleast one uncertainty comprises assigning a probabilistic distributionfunction to the at least one uncertainty.
 3. The method of claim 1,wherein obtaining, for each of a plurality of individuals, a geometricmodel comprises: splitting, for each of the plurality of individuals,the geometric model into a plurality of regions; and determining, foreach of the plurality of individuals, the sensitivity of the blood flowcharacteristic to uncertainty in features of the plurality of regions bysolving a stochastic algorithm for each of the plurality of regions ofthe geometric model.
 4. The method of claim 3, wherein solving thestochastic algorithm comprises utilizing quadrature points identified bya Smolyak sparse grid algorithm.
 5. The method of claim 3, wherein aresolution of the sensitivity may be increased by splitting an increasednumber of the plurality of regions of the geometric model.
 6. The methodof claim 1, further comprising: constructing a machine-learningpredictor based on the sensitivities of the plurality of individuals,wherein the sensitivity of the blood flow characteristic of the patientis determined based on the machine-learning predictor.
 7. The method ofclaim 6, wherein constructing the predictor comprises splitting dataassociated with the sensitivities of the plurality of individuals into atraining set and a test set.
 8. A system for determining a sensitivityof a patient's blood flow characteristic to uncertainty in one or moreanatomical or geometrical features of the patient, the systemcomprising: a data storage device storing instructions for determiningsensitivity; and a processor configured to execute the instructions toperform a method including the steps of: obtaining, for each of aplurality of individuals, a geometric model and a sensitivity of a bloodflow characteristic to at least one uncertainty, wherein the at leastone uncertainty comprises one or more uncertain parameters, uncertainclinical variables, and/or uncertain geometries; obtaining, for apatient, a geometric model of at least part of the patient's vascularsystem, a designation of a blood flow characteristic, and a value of theat least one uncertainty for at least part of the patient's vascularsystem; and determining a sensitivity of the blood flow characteristicof the patient for the at least the part of the patient's vascularsystem, based on one or more sensitivities of the plurality ofindividuals.
 9. The system of claim 8, wherein obtaining a sensitivityof a blood flow characteristic to the at least one uncertainty comprisesassigning a probabilistic distribution function to the at least oneuncertainty.
 10. The system of claim 8, wherein obtaining, for each of aplurality of individuals, a geometric model comprises: splitting, foreach of the plurality of individuals, the geometric model into aplurality of regions; and determining, for each of the plurality ofindividuals, the sensitivity of the blood flow characteristic touncertainty in features of the plurality of regions by solving astochastic algorithm for each of the plurality of regions of thegeometric model.
 11. The system of claim 10, wherein solving thestochastic algorithm comprises utilizing quadrature points identified bya Smolyak sparse grid algorithm.
 12. The system of claim 10, wherein aresolution of the sensitivity may be increased by splitting an increasednumber of the plurality of regions of the geometric model.
 13. Thesystem of claim 8, wherein the processor is further configured for:constructing a machine-learning predictor based on the sensitivities ofthe plurality of individuals, wherein the sensitivity of the blood flowcharacteristic of the patient is determined based on themachine-learning predictor.
 14. The system of claim 13, whereinconstructing the predictor comprises splitting data associated with thesensitivities of the plurality of individuals into a training set and atest set.
 15. A non-transitory computer readable medium for use on atleast one computer system containing computer-executable programminginstructions for determining a sensitivity of a patient's blood flowcharacteristic to uncertainty in one or more anatomical or geometricalfeatures of the patient, the method comprising: obtaining, for each of aplurality of individuals, a geometric model and a sensitivity of a bloodflow characteristic to at least one uncertainty, wherein the at leastone uncertainty comprises one or more uncertain parameters, uncertainclinical variables, and/or uncertain geometries; obtaining, for apatient, a geometric model of at least part of the patient's vascularsystem, a designation of a blood flow characteristic, and a value of theat least one uncertainty for at least part of the patient's vascularsystem; and determining a sensitivity of the blood flow characteristicof the patient for at least the part of the patient's vascular system,based on one or more sensitivities of the plurality of individuals. 16.The computer readable medium of claim 15, wherein obtaining asensitivity of a blood flow characteristic to the at least oneuncertainty comprises assigning a probabilistic distribution function tothe at least one uncertainty.
 17. The computer readable medium of claim15, wherein obtaining, for each of a plurality of individuals, ageometric model comprises: splitting, for each of the plurality ofindividuals, the geometric model into a plurality of regions; anddetermining, for each of the plurality of individuals, the sensitivityof the blood flow characteristic to uncertainty in features of theplurality of regions by solving a stochastic algorithm for each of theplurality of regions of the geometric model.
 18. The computer readablemedium of claim 17, wherein solving the stochastic algorithm comprisesutilizing quadrature points identified by a Smolyak sparse gridalgorithm.
 19. The computer readable medium of claim 17, wherein aresolution of the sensitivity may be increased by splitting an increasednumber of the plurality of regions of the geometric model.
 20. Thecomputer readable medium of claim 15, the method further including thesteps of: constructing a machine-learning predictor based on thesensitivities of the plurality of individuals, wherein the sensitivityof the blood flow characteristic of the patient is determined based onthe machine-learning predictor.